Use 3D vectors with dot and cross products to find angles, projections, lines and planes.

What this dot point is asking

In three dimensions a vector has three components, $\mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k}$, often written as a column $(a_1, a_2, a_3)$. The basic operations carry over from two dimensions, but two new ideas become essential: the cross product, and the description of lines and planes.

Magnitude and unit vectors

The magnitude (length) is

A unit vector in the direction of $\mathbf{a}$ is $\hat{\mathbf{a}} = \dfrac{\mathbf{a}}{|\mathbf{a}|}$. Unit vectors are useful whenever you only care about direction, for example when writing the direction of a line.

The dot (scalar) product

where $\theta$ is the angle between the vectors. Rearranging gives the angle:

Two non-zero vectors are perpendicular exactly when $\mathbf{a}\cdot\mathbf{b} = 0$. The scalar projection of $\mathbf{a}$ onto $\mathbf{b}$ is $\dfrac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{b}|}$, and the vector projection is $\dfrac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{b}|^2}\,\mathbf{b}$.

The cross (vector) product

The cross product produces a vector perpendicular to both inputs:

Its magnitude is $|\mathbf{a}\times\mathbf{b}| = |\mathbf{a}|\,|\mathbf{b}|\sin\theta$, which equals the area of the parallelogram spanned by $\mathbf{a}$ and $\mathbf{b}$. Note that $\mathbf{a}\times\mathbf{b} = -(\mathbf{b}\times\mathbf{a})$, so order matters.

Lines and planes

The vector equation of a line through point $\mathbf{r}_0$ with direction $\mathbf{d}$ is

The equation of a plane with normal $\mathbf{n} = (a,b,c)$ through point $\mathbf{r}_0$ is $\mathbf{n}\cdot(\mathbf{r} - \mathbf{r}_0) = 0$, which expands to the Cartesian form $ax + by + cz = d$ where $d = \mathbf{n}\cdot\mathbf{r}_0$. To find a normal to a plane through three points, take the cross product of two vectors lying in the plane.

Putting it together

Many exam questions chain these tools: find a direction with subtraction, a normal with a cross product, an angle with a dot product, then assemble a line or plane equation. Lay your working out in clear named steps so each result is easy to reuse later in the question.